Star Covers and Star Partitions of Cographs and Butterfly-free Graphs

A graph that is isomorphic to K-1,K- r for some r >= 0 is called a star. For a graph G = (V, E), any subset S of its vertex set V is called a star of G if the subgraph induced by S is a star. A collection C = {V-1, . . . , V-k} of stars in G is called a star cover of G if V-1 boolean OR. . . boolean OR V-k = V. A star cover C of G is called a star partition of G if it is also a partition of V. Given a graph G, the problem STAR COVER asks for a star cover of G of minimum size. Given a graph G, the problem STAR PARTITION asks for a STAR PARTITION of G of minimum size. Both the problems are NP-hard even for bipartite graphs [24]. In this paper, we obtain exact O(n(2)) time algorithms for both STAR COVER and STAR PARTITION on (C-4, P-4)-free graphs and on (2K(2), P-4)-free graphs. We also prove that STAR COVER and STAR PARTITION are polynomially equivalent, up to the optimum value, for butterfly-free graphs and present an O(n(14)) time O(log n)-approximation algorithm for these equivalent problems on butterfly-free graphs. We also obtain O(log n)-approximation algorithms for STAR COVER on hereditary graph classes.